Question 1

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. In a random sample of 200 patients, 30 reported side effects. Construct a 95% confidence interval for the true proportion of patients experiencing side effects. Which of the following is the correct formula for the lower bound of the confidence interval?

Accepted Answer

$$\hat{p} - z_{\alpha/2} \cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 2

Which of the following is NOT an assumption required for constructing a one-sample t confidence interval for the mean?

Accepted Answer

The sample size must be less than 5% of the population size.

Question 3

A researcher is testing whether a new teaching method improves test scores. The null hypothesis is $$H_0$$: \mu = 75$$ and the alternative is H_1$: \mu \u003e 75. $$Which type of test is this?

Accepted Answer

Right-tailed t test

Question 4

In a study comparing two diets, 40 out of 100 people on Diet A lost weight, while 30 out of 100 people on Diet B lost weight. What is the pooled sample proportion $$\hat{p}$$ used in the two-sample z test for proportions?

Accepted Answer

$$\dfrac{40 + 30}{100 + 100}$$

Question 5

Which of the following best describes a Type II error in hypothesis testing?

Accepted Answer

Failing to reject the null hypothesis when the alternative hypothesis is true.

Question 6

A hospital administrator wants to estimate the average wait time in the emergency room. She collects a random sample of 50 wait times, with a sample mean of 2.5 hours and a sample standard deviation of 0.8 hours. What is the correct formula for the 95% confidence interval for the mean wait time?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}}$$

Question 7

Which of the following scenarios would require the use of a paired (dependent samples) t test?

Accepted Answer

Measuring blood pressure before and after administering a drug to the same group of patients.

Question 8

A researcher wants to determine the sample size needed to estimate a population proportion with a margin of error of 0.03 at a 95% confidence level. No prior estimate is available. Which formula should be used?

Accepted Answer

$$n = 0.25 \left( \dfrac{z_{\alpha/2}}{E} \$$right)^2$$

Question 9

Which of the following statements about the bootstrap method is TRUE?

Accepted Answer

It involves resampling with replacement from the observed data to estimate the sampling distribution of a statistic.

Question 10

A two-sample t test for independent means (assuming unequal variances) is being conducted. Which of the following is the correct formula for the test statistic?

Accepted Answer

$$t_0$$ = \dfrac{(\bar{x}_1 - \bar{x}_2) - (\$$mu_1$$ - \$$mu_2$$)}{\sqrt{\dfrac{$$s_1^2$$}{$$n_1$$} + \dfrac{$$s_2^2$$}{$$n_2$$}}}$$

Question 11

A clinical trial compares the effectiveness of two medications. The 95% confidence interval for the difference in mean recovery times (Medication A minus Medication B) is $$(-2.1, 0.5). $$What can be concluded at the 5% significance level?

Accepted Answer

There is no statistically significant difference in mean recovery times between the two medications.

Question 12

Which of the following is the correct formula for the margin of error in a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}}$$

Question 13

A random sample of 400 voters shows that 220 support a new policy. Test the hypothesis $$H_0$$: p = 0.5$$ vs H_1$: p 
eq 0.5 at $$the 5% significance level. Which formula should be used to calculate the test statistic?

Accepted Answer

$$z_0$$ = \dfrac{\hat{p} - $$p_0$$}{\sqrt{\dfrac{$$p_0(1$$-$$p_0$$)}{n}}}$$

Question 14

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we were to take many random samples and construct a confidence interval from each, about 95% of those intervals would contain the true population mean.

Question 15

A scientist wants to compare the mean cholesterol levels of two independent groups: one taking a new drug and one taking a placebo. The sample sizes are 25 and 30, and the sample variances are not equal. Which test should be used?

Accepted Answer

Two-sample t test for independent means (assuming unequal variances)

Question 16

Which of the following is a requirement for using the normal approximation in a one-sample z test for proportions?

Accepted Answer

$$n p_0$ (1 - p_0$) \geq 10$$

Question 17

A company wants to estimate the mean time it takes to assemble a product. The desired margin of error is 2 minutes, the estimated standard deviation is 6 minutes, and a 95% confidence level is required. Which formula should be used to determine the minimum sample size?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \cdot s}{E} \$$right)^2$$

Question 18

Which of the following best describes the difference between statistical significance and practical significance?

Accepted Answer

Statistical significance means the result is unlikely due to chance, while practical significance means the result is large enough to be meaningful in real life.

Question 19

A randomization test is most appropriate in which of the following situations?

Accepted Answer

When the assumptions of traditional parametric tests are not met.

Statistics Unit 3: Confidence Intervals and Hypothesis Testing (Chapters 9-11) – Study Guide

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